Analytical Solutions of Laminar and Turbulent Dam Break Wave H. Chanson Civil Engineering, The University of Queensland, Brisbane, Australia ABSTRACT: Modern predictions of dam break wave rely often on numerical predictions validated with limited data sets. Basic theoretical studies were rare during the past decades. Herein simple solutions of the dam break wave are developed using the Saint-Venant equations for an instantaneous dam break with a semiinfinite reservoir in a wide rectangular channel initially dry. New analytical equations are obtained for both turbulent flow and laminar flow motion on horizontal and sloping inverts with non-constant friction factor. The results are validated by successful comparisons between theoretical results and several experimental data sets. The theoretical developments yield simple explicit analytical expressions of the dam break wave with flow resistance. The results compare well with experimental data and more advanced theoretical solutions. The developments are simple, yielding nice pedagogical applications of the method of characteristics for horizontal and sloping channels. These analytical solutions may be used to validate numerical solutions, while the simplicity of the equations allows some extension to more complex fluid flows (e.g. non-Newtonian fluids). 1 INTRODUCTION Dam break waves have been responsible for numerous losses of life (e.g. Fig. 1). Figure 1 illustrates two tragic accidents. Related situations include flash flood runoff in ephemeral streams, debris flow surges and tsunami runup on dry coastal plains. In all cases, the surge front is a sudden discontinuity characterised by extremely rapid variations of flow depth and velocity. Dam failures motivated basic studies on dam break wave, including the milestone contribution by Ritter (1892) following the South Fork (Johnstown) dam disaster (USA, 1889). Physical modelling of dam break wave is relatively limited despite a few basic experiments (Table 1). In retrospect, the experiments of Schoklitsch (1917) were well ahead of their time, and demonstrated that Armin von Schoklitsch (1888–1968) had a solid understanding of both physical modelling and dam break processes. For the last 40 years, there have been substantial efforts in dam break research, in particular with the European programs CADAM and IMPACT, and some American programs. These efforts were associated with the development of numerous numerical models and a few physical model studies. But, at a few exceptions, there has been no new theoretical developments since the basic works of Dressler (1952) and Whitham (1955) for horizontal channel and Hunt (1982,1984) for sloping channel. There have been also contradictory arguments on flow fundamentals. For example, some measurements highlighted a boundary layer region in the surging wave leading edge (e.g. Mano 1984, Davies 1988, Fujima and Shuto 1990, Chanson 2005c) while others obtained quasi-ideal vertical velocity distributions (Estrade 1967, Wang 2002, Jensen et al. 2003). Most studies were restricted to Newtonian fluids, but a few (e.g. Piau 1996, Chanson et al. 2006), while air entrainment at the shock leading edge was nearly always ignored (Chanson 2004a). This study describes simple analytical solutions for instantaneous dam break wave based upon the method of characteristics. The solutions are developed for initially-dry horizontal and sloping channels with either laminar or turbulent motion assuming non-constant friction coefficient. Results are compared with several data sets obtained in largesize facilities. It is the aim of this work to provide simple explicit solutions of dam break wave problems that are easily understood by students, young researchers and professionals. 2 DAM BREAK WAVE ON A HORIZONTAL CHANNEL 2.1 Presentation A dam break wave is the flow resulting from a sudden release of a mass of fluid in a channel. For onedimensional applications, the continuity and momentum equations yield the Saint-Venant equations: (A) St Francis dam (USA) in March 1928 shortly after failure (Courtesy of Santa Clarita Valley Historical Society) - Looking downstream at the remnant piece ∂d V ⎛ ∂A ⎞ ∂d A ∂V + *⎜ ⎟ +V * + * ∂x B ⎝ ∂x ⎠ d =cons tan t ∂t B ∂x ∂V ∂V ∂d +V * + g * + g *( S f − So ) = 0 ∂t ∂x ∂x =0 (1) (2) where d is the water depth, V is the flow velocity, t is the time, x is the horizontal co-ordinate with x = 0 at the dam, So is the bed slope (So = sinθ), θ is the angle between invert and horizontal with θ > 0 for downward slopes, and Sf is the friction slope (Liggett 1994, Viollet et al. 2002, Chanson 2004b). The Saint-Venant equations cannot be solved analytically usually because of non-linear terms that include the friction slope: Sf = (B) Malpasset dam (France) in 2004 looking at the right abutment (Courtesy of Sylvia Briechle) - Failure on 2 Dec. 1959 around 9:10pm Fig. 1 - Photographs of dam break accidents Table 1. Experimental studies of dam break wave _______________________________________________ Reference Slope Experimental configuration deg. _______________________________________________ Turbulent flows Schoklitsch 0 D ≤ 0.25 m, W = 0.6 m (1917) D ≤ 1 m, W = 1.3 m Dressler 0 D = 0.055 to 0.22 m, W = 0.225 m (1954) Smooth invert, Sand paper, Slats Cavaillé 0 L = 18 m, W = 0.25 m (1965) D = 0.115 to 0.23 m, Smooth invert D = 0.23 m, Rough invert Montuori 0.06 Casino I powerplant raceway (1965) L = 13.6 km, W = 1.95 m Estrade 0 L = 13.65 m, W =0.50 m (1967) D = 0.2 & 0.4 m, Smooth & Mortar L = 0.70 m, W = 0.25 m D = 0.3 m, Smooth & Rough invert Lauber 0 L < 3.6 m, W = 0.5 m, D < 0.6 m (1997) Smooth PVC invert _______________________________________________ Laminar flows Debiane 0 L < 0.66 m, W = 0.3 m, D = 0.055 m (2000) Glucose syrups (12 ≤ µ ≤ 170 Pa.s) _______________________________________________ Thixotropic flows Chanson et 15 L = 2 m, W = 0.34 m, D < 0.08 m al. (2006) Bentonite suspensions. ______________________________________________ * D : initial reservoir height; L : channel length; W : channel width. f V2 * 2 g * DH (3) where DH is the hydraulic diameter and f is the Darcy-Weisbach friction factor that is a non linear function of Reynolds number and relative roughness. For a prismatic rectangular channel, Equations (1) and (2) yield: ∂d ∂ (d *V ) + = 0 ∂t ∂x (4) ∂V ∂V ∂d f V 2 +V * + g* + * = g * So ∂t ∂x ∂x 2 DH (5) 2.1.1 Ideal fluid flow solution For a frictionless dam break in a wide horizontal channel, the analytical solution of Equations (4) and (5) yields Ritter solution (Ritter 1892) : U = 2 g*D (6) x d = 2 − 3* D t* g*D (7) where U is the wave front celerity (Fig. 2). Equation (7) was first derived by Barré de Saint Venant (1871). Equations (6) and (7) give the wave front celerity and instantaneous free-surface profile at any instant t > 0. The flow velocity everywhere may be derived from the method of characteristics. xs ∫ d * dx = − x 2 *D at any time t (9) x2 2.1.2 Real fluid flow solution Let us consider an instantaneous dam break in a rectangular, prismatic channel with bed friction and for a semi-infinite reservoir. The turbulent dam break flow is analysed as an ideal-fluid flow region behind a flow resistance-dominated tip zone (Fig. 2). Whitham (1955) introduced this conceptual approach that was used later by other researchers (e.g. Piau 1996, Debiane 2000), but these mathematical developments differ from the present simple solution. In Equation (9), right handside term, the negative sign indicates that the negative wave propagates upstream (Fig. 2) and that x2 < 0. 2.2 Solution for turbulent flow motion (So = 0) For turbulent motion, the flow resistance may be approximated by the Altsul formula first proposed in 1952 : 1/ 4 ⎛ k 100 ⎞ f = 0.1* ⎜ 1.46* s + ⎟ DH Re ⎠ ⎝ (10) where ks is the equivalent sand roughness height, Re is the Reynolds number (Re = ρ V DH/µ), and ρ and µ are the fluid density and dynamic viscosity respectively. Equation (10) is a simplified expression that satisfies the Moody diagram and which may be used to initialise the calculations with the ColebrookWhite formula (Idelchik 1969, Chanson 2004b). Assuming that Equation (10) holds in unsteady flows, and for a wide channel (i.e. DH ≈ 4 d), it may be rewritten as : f = 1 ⎛d⎞ ⎜ ⎟ ⎝D⎠ 1/ 4 * G1/ 4 (10b) where the dimensionless term G equals : Fig. 2 - Sketch of dam break wave in a horizontal channel In the wave tip region (x1 ≤ x ≤ xs, Fig. 2), the flow velocity does not vary rapidly in the wave tip zone (Dressler 1954, Estrade 1967, Liem and Kongeter 1999). If the flow resistance is dominant, and the acceleration and inertial terms are small, the dynamic wave equation (Eq. (5)) may be reduced into a diffusive wave equation. Assuming that the flow velocity in the tip region is about the wave front celerity U, Equation (5) yields : ∂d f U 2 = 0 + * ∂x 8 g * d Wave tip region (8) In the ideal fluid flow region behind the frictiondominated tip region, Equations (4) and (5) are used with f = 0. Chanson (2005a,b) solved analytically Equations (4), (5) and (8) assuming a constant Darcy friction factor in the wave tip region. In the following paragraphs, the method will be extended to non-constant friction factor. The solutions involve the conservation of mass. That is, the mass of fluid in motion (x2 ≤ x ≤ xs) must equal the initial mass of fluid at rest (x2 ≤ x ≤ 0): G = 3.65E − 5 * ks + D 2.5E − 3 U Re D * g*D (11) and ReD = ρ g D3/µ is a dimensionless Reynolds number defined in terms of the initial reservoir height D and fluid properties. The integration of Equation (9) gives the shape of the wave front : 4/9 2 ⎛ U ⎞ x s − x ⎞⎟ d ⎛⎜ 9 1/ 4 ⎟ * = *G *⎜ ⎜ g*D ⎟ D ⎟⎟ D ⎜⎜ 32 ⎠ ⎝ ⎝ ⎠ Wave tip region (12) for turbulent dam break wave. Combining Equations (4), (5), (8), (9) and (12), the flow properties at the transition between the wave tip and ideal-fluid regions are: 2 ⎛ U ⎞ x s − x1 ⎞⎟ d1 ⎛⎜ 9 1/ 4 ⎟ * *G *⎜ = ⎜ g*D ⎟ D ⎜⎜ 32 D ⎟⎟ ⎠ ⎝ ⎝ ⎠ 4/9 (13) ⎛3 ⎞ x1 U =⎜ * − 1⎟ g * D * t ⎜⎝ 2 g * D ⎟⎠ (14) The conservation of mass (Eq. (9)) yields an explicit relationship between the wave celerity U and time t : front celerity or location data are approximate and often unsuitable. Present results were also compared successfully with the analytical solutions of Dressler (1952), Whitham (1954) and Chanson (2005b). The differences were small and much smaller than experimental uncertainties. 7/2 ⎛ 1 ⎞ ⎜1 − * U ⎟ g * D ⎟⎠ 32 ⎜⎝ 2 * 2 13 ⎛ ⎞ U ⎟ G1/ 4 * ⎜ ⎜ g*D ⎟ ⎝ ⎠ 1.2 = (15) ⎛ G1/ 4 * ⎜ ⎜ ⎝ t.sqrt(g/D)=56.8 1 t.sqrt(g/D)=83.6 Ritter solution 0.8 Theory t.sqrt(g/D)=21.5 Theory t.sqrt(g/D)=57 0.6 Theory t.sqrt(g/D)=84 0.4 ⎞ xs ⎛ 3 U g *t =⎜ * − 1⎟ * D ⎜⎝ 2 g * D ⎟⎠ D 1 t.sqrt(g/D)=21.5 t.sqrt(g/D)=52.2 g *t D as well as the location of the wave leading edge: 32 * + 9 d/D 0.2 ⎛ 1 U ⎞⎟ * ⎜1 − * 2 ⎜ g * D ⎟⎠ U ⎞⎟ ⎝ 2 g * D ⎟⎠ 9/2 0 (16) -1.4 -0.9 ⎞ d 1 ⎛ x = *⎜ 2 − ⎟ D 9 ⎜⎝ t * g * D ⎟⎠ 1d/D 0.6 1.1 Ritter solution t.sqrt(g/D)=13 0.8 t.sqrt(g/D)=64 DRESSLER, Slats, t.sqrt(g/D)=13 x ≤ x2 (17a) 0.6 DRESSLER, Slats, t.sqrt(g/D)=64 0.4 2 x2 ≤ x ≤ x1 (17b) 0.2 4/9 2 ⎞ ⎛ ⎞ d ⎛⎜ 9 U ⎟ * xs − x ⎟ = ⎜ * G1/ 4 * ⎜ ⎜ g*D ⎟ D ⎜ 32 D ⎟⎟ ⎝ ⎠ ⎝ ⎠ x1 ≤ x ≤ xs (17c) d =0 D x/t.sqrt(g.D) 0.1 (A) Experiments of Cavaillé (1965) with smooth invert The instantaneous free-surface profile is then : d =1 D -0.4 xs ≤ x (17d) where G is defined in Equation (11). Equation (15) gives the wave front celerity at any instant t (t > 0), Equations (16) and (17) provide the entire free-surface profile, and the flow velocity may be deduced everywhere. The analytical solution was compared systematically with several experimental studies (Table 1). Figure 3 shows some examples illustrating the good agreement between theory and experiments. Figure 3 shows instantaneous free-surface profiles for different types of invert and different times from dam break. The comparative analysis suggested that the results were sensitive to the choice of the equivalent sand roughness height ks and that its selection must be based upon a match with instantaneous freesurface profiles. Alternate comparisons with wave 0 -1 -0.5 0 x/(t.sqrt(g*D)) 0.5 1 1.5 2 (B) Experiments of Dressler (1954) with slats - Calculations assuming ks/D = 0.2 Fig. 3 - Instantaneous free-surface profiles : comparison between the present theory and experiments (turbulent flow motion) 2.3 Solution for laminar flow motion (So = 0) In very viscous fluids, the flow motion is laminar and the resistance must be estimated accordingly. Some researchers argued that the flow resistance in unsteady flows may differ from classical steady flow estimates and they suggested that the friction factor may be estimated as : f = α 64/Re where α is a correction coefficient and α = 1 for steady flows (e.g. Aguirre-Pe et al. 1995, Debiane 2000). For a wide channel (i.e. DH ≈ 4 d), the Darcy friction factor in the wave tip region becomes: f = 16* α U d * ReD * g*D D (18) The integration of Equation (8) yields the shape of wave tip region for a laminar flow motion: x − x⎞ µ *U d ⎛ ⎟ * s = ⎜⎜ 6 * α * 2 D ⎝ D ⎟⎠ ρ *g*D The instantaneous free-surface profile is : 1/ 3 (19) Equation (19) is compared with experimental data and with Equation (12) in Figure 4. Equation (19) was found to be in qualitative agreement with experimental data by Tinney and Bassett (1961), Aguirre-Pe et al. (1995) and Debiane (2000), and the analytical solution of Hunt (1994). Figure 4 presents the dimensionless water depth d/d1 as function of the dimensionless distance (xsx)/(xs-x1) where the subscripts s and 1 refer respectively to the wave leading edge and to the transition between ideal fluid and wave tip regions. Note the good agreement between Equations (12) and (19) and their corresponding experimental data. The results suggest further that the wave front is comparatively steeper in a laminar dam break wave than in turbulent flow motion. d/d1 1 CAVAILLE smooth invert, t.sqrt(g/D)=56.8 0.8 DRESSLER smooth invert, t.sqrt(g/D)=66 DRESSLER slats, t.sqrt(g/D)=13 0.6 Turbulent flow, ks/D=0.35, t.sqrt(g/D)=55.6 Turbulent flow 0.4 DEBIANE, ReD=4.7 t.sqrt(g/D)=5620 DEBIANE, ReD=0.67, t.sqrt(g/D)=8.6 0.2 Laminar flow motion 0 0.8 0.6 (xs0.4 -x)/(xs-x1)0.2 0 Fig. 4 - Dimensionless free-surface profiles in the wave tip region - Comparison between Equations (12) and (19), and experimental data (Laminar flow: Debiane 2000, Turbulent flow: Dressler 1954, Cavaillé 1965) For a rectangular horizontal channel, the integration of the continuity equation within the boundary conditions gives a simple expression of the wave front celerity : ⎛ 1 ⎞ ⎜1 − * U ⎟ ⎜ 2 g * D ⎟⎠ 1 * Re D * ⎝ U 8 *α g*D x ≤ x2 (22a) ⎞ d 1 ⎛ x = *⎜ 2 − ⎟ D 9 ⎜⎝ t * g * D ⎟⎠ 2 x2 ≤ x ≤ x1 (22b) x − x⎞ µ *U d ⎛ ⎟ * s = ⎜⎜ 6 * α * 2 D ⎝ D ⎟⎠ ρ *g*D d =0 D 1/ 3 x1 ≤ x ≤ xs (22c) xs ≤ x (22d) The characteristic locations x1 and x2 are given by: ⎛3 ⎞ x1 U =⎜ * − 1⎟ g * D * t ⎜⎝ 2 g * D ⎟⎠ CAVAILLE rough invert, t.sqrt(g/D)=55.6 Laminar flow 1 d =1 D x2 t* g*D (23) = −1 (24) A comparison with experimental results is restricted by a lack of detailed measurements. Debiane (2000) presented a rare set of experimental data performed in a 3 m long 0.3 m wide channel using glucose-syrup solutions of dynamic viscosities between 12 to 170 Pa.s. Figures 4 and 5 show some comparison between theory and experiments during the early stages of the dam break. Figure 5 presents a comparison in terms of dimensionless wave front location in horizontal channel. Experimental data are compared with Equation (21) for three values of α and with an ideal dam break (Ritter solution). 10 xs/D Ritter 5 = 1 g *t D Ritter solution (20) Theory (Alpha = 1) Theory (Alpha = 2) Theory (Alpha = 10) Data DEBIANE while the location of the wave leading edge is: 0.1 ⎞ xs ⎛ 3 U g *t =⎜ * − 1⎟ * D ⎜⎝ 2 g * D ⎟⎠ D 0.1 6 ⎛ 1 Re D U ⎞⎟ 1 + (21) * * ⎜1 − * ⎜ 2 U 6 *α g * D ⎟⎠ ⎝ g*D t.sqrt(g/D)1 10 Fig. 5 - Dimensionless wave front location in laminar dam break flow motion - Comparison between Equations (21), ideal fluid flow solution and experiments (Debiane 2000) Experiments with very viscous solutions showed a trend similar to theoretical calculations (Present study, Piau and Debiane 2005) (Fig. 4). But the data gave consistently a slower wave front propagation as observed in Figure 5. Debiane experienced some experimental difficulties (Debiane 2000, pp. 161-163). The gate opening was relatively slow and could not be considered to be an instantaneous dam break. Some fluid remained attached to the gate while other portions were projected away. Sidewall effects were also observed for the most viscous solutions (µ = 170 Pa.s) including some fluid crystallisation next to the walls at the shock front. Debiane's comments underlined the difficulties associated with physical modelling, but his advice is nonetheless useful for future experiments. Fig. 6 - Definition sketch of dam break wave in a sloping channel 3.2 Solution for turbulent flow motion 3 DAM BREAK WAVE ON A SLOPING CHANNEL Considering a turbulent flow down a constant slope, the momentum equation in the wave tip region becomes : 3.1 Presentation The above development were obtained for a horizontal channel (So = 0). They may be extended to situations with some initial flow motion and to sloping invert. The former development was discussed by Chanson (2005a) and the latter case is discussed below. For an ideal fluid flow, the friction slope is zero and Ritter solution may be extended using a method of superposition (Peregrine and Williams 2001, Chanson 2005a). For a mild, constant slope, the solution of the Saint Venant equations is : 2 ⎞ d 1 ⎛ 1 g x = *⎜ 2 + * * So * t − ⎟ D 9 ⎜⎝ 2 D t * g * D ⎟⎠ x, 1 g -1 ≤ ≤ 2 + * So * * t (25) 2 D t* g*D ⎞ V 2 ⎛ g x = * ⎜1 + So * *t + ⎟ D g * D 3 ⎜⎝ t * g * D ⎟⎠ x 1 g -1 ≤ * t (26) ≤ 2 + * So * 2 D t* g*D where the bed slope So is positive for a downwards lope. The celerity of the wave leading edge is : U g = 2 + So *t D g*D (27) For a channel inclined upwards (e.g. beach face), the wave front stops (U = 0) at : t = -2 D/g/So. ∂d f U 2 + * − So = 0 ∂x 8 g * d Wave tip region (28) where the Darcy friction factor f satisfies Equation (10). The complete integration yields a complicated result that is presented in the Appendix. A Taylor series expansion of the solution gives the earlier result: 4/9 2 ⎛ ⎞ x − x ⎞⎟ d ⎛⎜ 9 U ⎟ * s = ⎜ * G1/ 4 * ⎜ ⎜ g*D ⎟ D ⎜ 32 D ⎟⎟ ⎝ ⎠ ⎝ ⎠ Wave tip region (12) where the dimensionless term G is : G = 3.65E − 5 * ks + D 2.5E − 3 U Re D * g*D (11) and ReD = ρ g D3/µ. Equation (12) implies that both acceleration, inertial and gravity terms are small in the wave tip region. The assumption is valid on mild slope and more generally for Sf >> So. Equation (12) was successfully compared with field observations in the Casino I powerplant tunnel by Montuori(1965). Using Equation (12), the integration of the conservation of mass (Eq. (9)) yields an explicit relationship between the wave front celerity U and the time since dam break t : ⎛ 1 ⎞ ⎜1 + * g * S o * t − 1 * U ⎟ D 2 g * D ⎟⎠ 32 ⎜⎝ 2 * 2 13 ⎛ U ⎞ 1/ 4 ⎟ G *⎜ ⎜ g*D ⎟ ⎝ ⎠ = 7/2 g * t (29) D while the location of the wave front is: ⎛3 ⎞ xs U g g − * S o * t − 1⎟ * *t = ⎜ * ⎜ ⎟ D D g*D ⎝2 ⎠ D ⎞ xs ⎛ 3 U g g * S o * t − 1⎟ * *t =⎜ * − ⎜ ⎟ D ⎝2 D g*D ⎠ D ⎛ 1 ⎞ ⎜1 + * g * S o * t − 1 * U ⎟ D 2 g * D ⎟⎠ 32 ⎜⎝ 2 + * 2 9 ⎛ ⎞ U ⎟ G1/ 4 * ⎜ ⎜ g*D ⎟ ⎝ ⎠ 4 9/ 2 (30) The instantaneous free-surface profile at any instant t > 0 is : d =1 D x ≤ x2 (31a) ⎛ 1 4 1 g 1 U ⎞ + * * ⎜1 + * * So * t − * ⎟ (34) 2 ⎜ 2 U 2 f D g * D ⎟⎠ ⎝ g*D The instantaneous free-surface equations are : d =1 D x ≤ x2 (35a) 2 ⎞ d 1 ⎛ 1 g x ⎟ * So * t − = * ⎜⎜ 2 + * D 9 ⎝ 2 D d * D * t ⎟⎠ x2 ≤ x ≤ x1 (35b) 2 ⎞ d 1 ⎛ 1 g x ⎟ * So * t − = * ⎜⎜ 2 + * D 9 ⎝ 2 D d * D * t ⎟⎠ x2 ≤ x ≤ x1 (31b) 4/9 2 ⎛ ⎛ U ⎞ x s − x ⎞⎟ d ⎜ 9 1/ 4 ⎟ * = *G *⎜ ⎜ g*D ⎟ D ⎟⎟ D ⎜⎜ 32 ⎝ ⎠ ⎝ ⎠ x1 ≤ x ≤ xs (31c) d =0 D xs ≤ x (31d) ⎞ g g * S o * t − 1⎟ * *t ⎟ D D ⎠ d =0 D 1/ 3 x1 ≤ x ≤ xs (35c) xs ≤ x (35d) The location of the transition between ideal fluid and wave tip regions is : x1 ⎛⎜ 3 U * = − ⎜ D ⎝2 g*D ⎞ g g * S o * t − 1⎟ * *t ⎟ D D ⎠ (36) Note also that the expression of the wave front celerity (Eq. (33)) is a non-linear equation. while the characteristic location x1 satisfies : x1 ⎛⎜ 3 U * = − ⎜ D ⎝2 g*D x − x⎞ µ *U d ⎛ ⎟ * s = ⎜⎜ 6 * α * 2 D ⎟⎠ D ⎝ ρ *g*D (32) Equations (29), (30), (31) and (32) define the entire free-surface at a given instant. Note that, in Equation (29), the time t appears in both right and left handside terms. Equation (29) is hence nonlinear. 3.4 Remarks The writer was alerted that Takahashi (1991) and Toro (2001) presented some related developments for dam break wave down a sloping chute. For a constant slope, Toro (2001) proposed a change of variables similar to that used by Peregrine and Williams (2001). 3.3 Solution for laminar flow motion The same reasoning may apply to a laminar flow motion. The exact solution in terms of wave front celerity is : ⎛ 1 g 1 U ⎞ + − 1 * * S * t * ⎜ ⎟ o 2 g * D ⎟⎠ ρ * g * D3 ⎜⎝ 2 D 1 * * U µ 8* α g*D = The wave front location is given by : g *t D 5 (33) 4 DISCUSSION The present theoretical solutions are based upon a few key assumptions. These include (a) V(x,t) = U(t) in the wave tip region and (b) some discontinuity in terms of ∂d/∂x, ∂V/∂x and bed shear stress at the transition between wave tip and ideal fluid flow regions (i.e. x = x1). Turbulent flow data tended to support the first approximation (a). For example, the data of Estrade (1967), Lauber (1997) and Liem and Kongeter (1999). The experimental results showed little longitudinal variations in velocity in the wave tip region. Further the comparisons between present diffusive wave solutions and experimental results were successful for a fairly wide range of experimental data obtained independently (Fig. 4). Such comparisons constitute a solid validation. Whitham (1955) and Dressler (1952) produced two very different analytical solutions with and without a discontinuity at the transition between ideal flow region and wave tip region (e.g. Fig. 7) Figure 7 illustrates differences in longitudinal bed shear stress distributions between the different theoretical models. Yet all theories gave close results in terms of wave front celerity and instantaneous free-surface profiles and good agreement with experimental data. All these suggest little effects of some discontinuity at x = x1, although the physics of a constant Darcy friction factor is questionable ! Fig. 7 - Sketch of longitudinal bed shear stress in turbulent dam break wave for the theoretical models of Dressler (1952), Whitham (1955), Chanson (2005a,b) and present study Ultimately, a practical question is : which is the best theoretical model for turbulent dam break wave? Dressler's (1952) method is robust, but the treatment of the wave leading edge is approximate : "To handle the tip region accurately, some type of boundary-layer technique would be necessary [...] but no results are yet available. [...] In the absence of more satisfactory boundary layer results, we will apply [...] approximate considerations to obtain some data about the wavefront" (Dressler 1952, pp. 223-224). Whitham's (1955) method is also a robust technique, but the results are asymptotic solutions. Both the model of Chanson (2005a.b) with constant friction factor and present results provide complete explicit solutions of the entire flow field. The velocity field and water depths are explicitly calculated everywhere. Note that the methods of Dressler, Whitham and Chanson (2005a,b) assume a constant friction factor, and the results are sensitive to the selection of the flow resistance coefficient value. In contrast, the present method requires only the selection of an equivalent roughness height. In practice, the selection of a suitable analytical model is linked with its main application. For example, for pedagogical purposes, the writer believes that Chanson's (2005a,b) model based upon a constant friction coefficient in the wave tip region is nicely suited to introductory courses and young professionals because of the explicit and linear nature of the results. The new developments shown herein present substantial advantages over previous works. The theoretical results yield simple explicit analytical expressions for real-fluid flows that compares well with experimental data and more advanced theoretical solutions. They cover a wide range of applications that include linear and turbulent flows, horizontal and sloping channels. The proposed solutions are simple pedagogical applications, linking together the ideal fluid flow equations yielding Ritter solution, with a diffusive wave equation for the wave tip region. These explicit results may be used to validate numerical solutions of the method of characteristics applied to dam break wave problem. The development may be also extended to a wider range of practical applications including non-Newtonian fluids (e.g. Chanson and Coussot 2005). 5 SUMMARY AND CONCLUSION New analytical solutions of dam break wave with bed friction are developed for turbulent and laminar flow motion with non constant friction factors. For each development, the dam break wave flow is analysed as a wave tip region where flow resistance is dominant, followed by an ideal-fluid flow region where inertial effects and gravity effects are dominant. In horizontal channels, turbulent flow solutions were validated by successful comparisons between several experimental data sets obtained in large-size facilities and theoretical results in terms of instantaneous free-surface profiles, wave front location and wave front celerity. The former comparison with instantaneous free-surface data is regarded as the most accurate technique to estimate the flow resistance and relative roughness, hence the best validation technique. A new analytical solution of dam break wave with laminar flow motion is also presented. The shape of the wave front compared well with experimental observations. The results were further extended to sloping channels. These simple theoretical developments yield explicit, non-linear expressions that compare well with experimental data. The simplicity of the equations will allow some extension of the method to more complex geometries, flow situations and broader applications, including non-Newtonian fluids, sheet flow and beach swash. It must be acknowledged that the present approach is limited to semi-infinite reservoir, rectangu- lar channel and quasi-instantaneous dam break. Further experimental data with very viscous fluids and non-Newtonian fluid flows are also needed. 6 APPENDIX Let us consider an instantaneous dam break wave in an initially dry inclined channel. In the frictiondominated tip region, the dynamic wave equation becomes: Wave tip region (A-1) where the Darcy friction factor satisfies the Altsul formula: 1 ⎛d⎞ ⎜ ⎟ ⎝D⎠ 1/ 4 * G1/ 4 (A-2) ⎛ 1 1⎞ ⎛ Χ ⎞ - ⎜⎜ − ⎟⎟ * ⎜⎜ ⎟⎟ ⎝ 5 5 ⎠ ⎝ So ⎠ ⎛ ⎛ Χ ⎞ 2 / 5 − 1 + 5 ⎛ Χ ⎞1 / 5 ⎞ Ln⎜ ⎜⎜ ⎟⎟ − * ⎜⎜ ⎟⎟ * d '1 / 4 + d ' ⎟ ⎜⎝ So ⎠ ⎟ 2 ⎝ So ⎠ ⎝ ⎠ 4/5 and the dimensionless term G is : G = 3.65 E − 5 * ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 4/5 ∂d f U 2 + * = So ∂x 8 g * d f = 4/5 2 * 10 − 2 * 5 ⎛ Χ ⎞ * ⎜⎜ ⎟⎟ 5 ⎝ So ⎠ 1/ 5 ⎛ ⎜ (−1 + 5 ) * ⎛⎜ Χ ⎞⎟ + 4 * d '1 / 4 ⎜S ⎟ ⎜ ⎝ o⎠ * ArcTan⎜ − 1/ 5 ⎜ ⎛Χ⎞ 10 − 2 * 5 * ⎜⎜ ⎟⎟ ⎜ ⎝ So ⎠ ⎝ ks + D 2.5E − 3 U Re D * g*D (A-3) ⎛ 1 1⎞ ⎛ Χ ⎞ + ⎜⎜ + ⎟⎟ * ⎜⎜ ⎟⎟ ⎝ 5 5 ⎠ ⎝ So ⎠ ⎛ ⎛ Χ ⎞ 2 / 5 1 + 5 ⎛ Χ ⎞1 / 5 ⎞ * Ln⎜ ⎜⎜ ⎟⎟ + * ⎜⎜ ⎟⎟ * d '1 / 4 + d ' ⎟ ⎜⎝ So ⎠ ⎟ 2 ⎝ So ⎠ ⎝ ⎠ x1 ≤ x ≤ xs (A-5) Equation (A-1) may be rewritten in dimensionless form as : ∂d ' X + 5/ 4 = So ∂x ' d ' (A-4) where d' = d/D, x' = x/D and 1/ 4 0.1 ⎛ 1.46 k s 100 ν ⎞ *⎜ * + * X= ⎟ 8 ⎝ 4 D 4 U *D ⎠ * U2 g*D with ν = µ/ρ is the kinematic viscosity. The boundary conditions are d' = 0 at the leading edge (x' = xs'). For a non-zero slope, the exact solution of Equation (A-4) is: 4/5 ⎛X⎞ So *( xs '− x ') = 0.020034* ⎜ ⎟ - d' S o ⎝ ⎠ 4/5 1/ 5 ⎛⎛ X ⎞ ⎞ 4 ⎛X⎞ − * ⎜ ⎟ * Ln ⎜ ⎜ ⎟ − d '1/ 4 ⎟ ⎜ ⎝ So ⎠ ⎟ 5 ⎝ So ⎠ ⎝ ⎠ 4/5 2 * 2 * (5 + 5 ) ⎛ Χ ⎞ * ⎜⎜ ⎟⎟ 5 ⎝ So ⎠ 1/ 5 ⎛ ⎜ (−1 + 5 ) * ⎛⎜ Χ ⎞⎟ + 4 * d '1 / 4 ⎜S ⎟ ⎜ ⎝ o⎠ * ArcTan⎜ − 1/ 5 ⎜ ⎛Χ⎞ 2 * (5 + 5 ) * ⎜⎜ ⎟⎟ ⎜ ⎝ So ⎠ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ REFERENCES Aguirre-Pe, J., Plachco, F.P., and Quisca, S. (1995). "Tests and Numerical One-Dimensional Modelling of a HighViscosity Fluid Dam-Break Wave." Jl of Hyd. Res., IAHR, Vol. 33, No. 1, pp. 17-26. Barré de Saint Venant, A.J.C. (1871). "Théorie et Equations Générales du Mouvement Non Permanent des Eaux Courantes." Comptes Rendus des séances de l'Académie des Sciences, Paris, France, Séance 17 July 1871, Vol. 73, pp. 147-154. Cavaillé, Y. (1965). "Contribution à l'Etude de l'Ecoulement Variable Accompagnant la Vidange Brusque d'une Retenue." Publ. Scient. et Techn. du Ministère de l'Air, No. 410, Paris, France, 165 pages. Chanson, H. (2004a). "Unsteady Air-Water Flow Measurements in Sudden Open Channel Flows." Experiments in Fluids, Vol. 37, No. 6, pp. 899-909. Chanson, H. (2004b). "The Hydraulics of Open Channel Flows: An Introduction." Butterworth-Heinemann, Oxford, UK, 2nd edition, 630 pages. Chanson, H. (2005a). "Applications of the Saint-Venant Equations and Method of Characteristics to the Dam Break Wave Problem." Report No. CH55/05, Dept. of Civil Engineering, The University of Queensland, Brisbane, Australia, May, 135 pages. Chanson, H. (2005b). "Analytical Solution of Dam Break Wave with Flow Resistance. Application to Tsunami Surges." Proc. 31th Biennial IAHR Congress, Seoul, Korea, B.H. Jun, S.I. Lee, I.W. Seo and G.W. Choi Editors, Theme D1, Paper 0137, pp. 3341-3353. Chanson, H. (2005c). "Air-Water and Momentum Exchanges in Unsteady Surging Waters : an Experimental Study." Experimental Thermal and Fluid Science, Vol. 30, No. 1, pp. 37-47. Chanson, H., and Coussot, P. (2005). "Fluid Dynamic Treatment of Thixotropic Debris Flows and Avalanches." Proc. 31th Biennial IAHR Congress, Seoul, Korea, B.H. JUN, S.I. LEE, I.W. SEO and G.W. CHOI Editors, Theme D.5, Paper 0039, pp. 5654-5666. Chanson, H., Jarny, S., and Coussot, P. (2006). "Dam Break Wave of Thixotropic Fluid." Jl of Hyd. Engrg., ASCE, Vol. 132, No. 3, pp. 280-293. Davies, T.R.H. (1988). "Debris Flow Surges - A Laboratory Investigation." Mitteilungen der Versuchsanstalt fur Wasserbau, Hydrologie und Glaziologie, No. 96, ETH-Zurich, Switzerland, 122 pages. Debiane, K. (2000). "Hydraulique des Ecoulements Laminaires à Surface Libre dans une Canal pour des Milieux Visqueux ou Viscoplastiques: Régimes Uniformes, Graduellement Varié, et Rupture de Barrage." Ph.D. thesis, University of Grenoble I, Rheology Laboratory INPG-UJF-CNRS, France, 273 pages. Dressler, R.F. (1952). "Hydraulic Resistance Effect upon the Dam-Break Functions." Jl of Research, Natl. Bureau of Standards, Vol. 49, No. 3, pp. 217-225. Dressler, R. (1954). "Comparison of Theories and Experiments for the Hydraulic Dam-Break Wave." Proc. Intl Assoc. of Scientific Hydrology Assemblée Générale, Rome, Italy, Vol. 3, No. 38, pp. 319-328. Estrade, J. (1967). "Contribution à l'Etude de la Suppression d'un Barrage. Phase Initiale de l'Ecoulement." Bulletin de la Direction des Etudes et Recherches, Series A, Nucléaire, Hydraulique et Thermique, EDF Chatou, France, No. 1, pp. 3-128. Fujima, K., and Shuto, N. (1990). "Formulation of Frictions Laws for Long Waves on a Smooth Dry Bed." Coastal Engineering in Japan, Vol. 33, No. 1, pp. 25-47. Hunt, B. (1982). "Asymptotic Solution for Dam-Break Problems." Jl of Hyd. Div., Proceedings, ASCE, Vol. 108, No. HY1, pp. 115-126. Hunt, B. (1984). "Perturbation Solution for Dam Break Floods." Jl of Hyd. Engrg., ASCE, Vol. 110, No. 8, pp. 1058-1071. Hunt, B. (1994). "Newtonian Fluid Mechanics Treatment of Debris Flows and Avalanches." Jl of Hyd. Engrg., ASCE, Vol. 120, No. 12, pp. 1350-1363. Idelchik, I.E. (1969). "Mémento des Pertes de Charge." Eyrolles Editor, Collection de la direction des études et recherches d'Electricité de France, Paris, France. Jensen, A., Pedersen, G.K., and Wood, D.J. (2003). "An Experimental Study of Wave Run-up on a Steep Beach." Jl of Fluid Mech., Vol. 486, pp. 166-188. Lauber, G. (1997). "Experimente zur Talsperrenbruchwelle im glatten geneigten Rechteckkanal." ('Dam Break Wave Experiments in Rectangular Channels.') Ph.D. thesis, VAWETH, Zürich, Switzerland. Liem, R., and Kongeter, J. (1999). "Application of High-Speed Digital Image Processing to Experiments on Dam Break Waves." Proc. CADAM Meeting, Zaragossa, Spain, pp. 399-411. Liggett, J.A. (1994). "Fluid Mechanics." McGraw-Hill, New York, USA. Mano, A. (1994). "Boundary Layer Developed near Surging Front." Coastal Engineering in Japan, Vol. 37, No. 1, pp. 23-39. Montuori, C. (1965). "Introduction d'un Débit Constant dans un Canal Vide." Proc. 11th IAHR Biennial Congress, Leningrad, Russia, Vol. 3, paper 3.55, pp. 1-7 &9 plates. Piau, J.M. (1996). "Flow of a Yield Stress Fluid in a Long Domain. Application to Flow on an Inclined Plane." Jl of Rheology, Vol. 40, No. 4, pp. 711-723. Piau, J.M., and Debiane, K. (2005). "Consistometers Rheometry of Power-Law Viscous Fluids." Jl of Non-Newtonian Fluid Mech., Vol. 127, pp. 213-224. Peregrine, D.H., and Williams, S.M. (2001). "Swash Overtopping a Truncated Plane Beach." Jl of Fluid Mech., Vol. 440, pp. 391-399. Ritter, A. (1892). "Die Fortpflanzung der Wasserwellen." Vereine Deutscher Ingenieure Zeitswchrift, Vol. 36, No. 2, 33, 13 Aug., pp. 947-954 (in German). Schoklitsch, A. (1917). "Über Dambruchwellen." Sitzungberichten der Königliche Akademie der Wissenschaften, Vienna, Vol. 126, Part IIa, pp. 1489-1514. Takahashi, T. (1991). "Debris Flow." IAHR Monograph, Balkema Publ., Rotterdam, The Netherlands. Tinney, E.R., and Bassett, D.L. (1961). "Terminal Shape of a Shallow Liquid Front." Jl of Hyd. Div., Proceedings, ASCE, Vol. 87, No. HY5, pp. 117-133. Toro, E.F. (2001). "Shock Capturing Methods for Free-Surface Shallow Flows." John Wiley, New York, USA, 309 pages. Viollet, P.L., Chabard, J.P., Esposito, P., and Laurence, D. (1998). "Mécanique des Fluides Appliquée. Ecoulements Incompressibles dans les Circuits, Canaux et Rivières, autour des Structures et dans l'Environnement." Presses des Ponts et Chaussées, Paris, France, 367 pages. Wang, Z.Y. (2002). "Initiation and Mechanism of Two Phase Debris Flow." Proc. Conf. on Flood Defence'2002, Ed. WU et al., Science Press, New York, pp. 1637-1648. Whitham, G.B. (1955). "The Effects of Hydraulic Resistance in the Dam-Break Problem." Proc. Roy. Soc. of London, Ser. A, Vol. 227, pp. 399-407. Internet resources "Fluid Mechanics and Hydraulic Engineering of Dam Break Wave. From the Dam Busters to Self-Flowing Concrete Testing." Internet resource. (http://www.uq.edu.au/~e2hchans/dam_brk.html) Photographs of dam break accidents (http://www.uq.edu.au/~e2hchans/photo.html) "Sabo check dams. Mountain protection systems in Japan." Internet resource. (http://www.uq.edu.au/~e2hchans/sabo.html)

© Copyright 2018 ExploreDoc